\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx\) [108]

Optimal result

Integrand size = 38, antiderivative size = 760 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{3 x^3}+\frac {B \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{2 x}-\frac {\sqrt {b^2-4 a c} (9 a B d+2 A (b d+a e)) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{6 \sqrt {2} a d \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} \left (4 A c d^2-3 a B d e-2 a A e^2+2 b d (3 B d+A e)\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{3 \sqrt {2} a d \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (B c d+b B e+2 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/3*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^3+1/2*B*(e*x^2+d)^(1/2)*(c* 
x^4+b*x^2+a)^(1/2)/x-1/12*(-4*a*c+b^2)^(1/2)*(9*B*a*d+2*A*(a*e+b*d))*(-a*( 
c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2 
*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b 
*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/a/d/(-a*(e+d/x^2)/((b+(-4*a 
*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/6*(-4*a*c+b^2)^(1/2 
)*(4*A*c*d^2-3*a*B*d*e-2*A*a*e^2+2*b*d*(A*e+3*B*d))*(-a*(c+a/x^4+b/x^2)/(- 
4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^ 
3*EllipticF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*( 
(-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/a/d/ 
(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*A*c*e+ 
B*b*e+B*c*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4 
*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a*c 
+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)),2^( 
1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))/(b+(-4 
*a*c+b^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^4,x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^4, x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4}dx\)

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^4,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ 
q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol 
yQ[Px, x]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{4}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{4}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="fricas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^4, x)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {d + e x^{2}} \sqrt {a + b x^{2} + c x^{4}}}{x^{4}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(c*x**4+b*x**2+a)**(1/2)/x**4,x)
 

Output:

Integral((A + B*x**2)*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{4}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^4, x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{4}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}\,\sqrt {c\,x^4+b\,x^2+a}}{x^4} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^4,x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=\text {too large to display} \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x)
 

Output:

( - sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a**2*b*e**2 - 2*sqrt(d + e* 
x**2)*sqrt(a + b*x**2 + c*x**4)*a**2*c*d*e + 2*sqrt(d + e*x**2)*sqrt(a + b 
*x**2 + c*x**4)*a**2*c*e**2*x**2 - 3*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c* 
x**4)*a*b**2*d*e + 2*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b**2*e** 
2*x**2 - 3*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*d**2 + 4*sqrt( 
d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*d*e*x**2 + 2*sqrt(d + e*x**2)* 
sqrt(a + b*x**2 + c*x**4)*b**3*d*e*x**2 + 2*sqrt(d + e*x**2)*sqrt(a + b*x* 
*2 + c*x**4)*b**2*c*d**2*x**2 - 4*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + 
c*x**4)*x**4)/(a**2*b*d*e**2 + a**2*b*e**3*x**2 + a**2*c*d**2*e + a**2*c*d 
*e**2*x**2 + a*b**2*d**2*e + 2*a*b**2*d*e**2*x**2 + a*b**2*e**3*x**4 + a*b 
*c*d**3 + 2*a*b*c*d**2*e*x**2 + 2*a*b*c*d*e**2*x**4 + a*b*c*e**3*x**6 + a* 
c**2*d**2*e*x**4 + a*c**2*d*e**2*x**6 + b**3*d**2*e*x**2 + b**3*d*e**2*x** 
4 + b**2*c*d**3*x**2 + 2*b**2*c*d**2*e*x**4 + b**2*c*d*e**2*x**6 + b*c**2* 
d**3*x**4 + b*c**2*d**2*e*x**6),x)*a**3*b*c**2*e**5*x**3 - 4*int((sqrt(d + 
 e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2*b*d*e**2 + a**2*b*e**3*x**2 
 + a**2*c*d**2*e + a**2*c*d*e**2*x**2 + a*b**2*d**2*e + 2*a*b**2*d*e**2*x* 
*2 + a*b**2*e**3*x**4 + a*b*c*d**3 + 2*a*b*c*d**2*e*x**2 + 2*a*b*c*d*e**2* 
x**4 + a*b*c*e**3*x**6 + a*c**2*d**2*e*x**4 + a*c**2*d*e**2*x**6 + b**3*d* 
*2*e*x**2 + b**3*d*e**2*x**4 + b**2*c*d**3*x**2 + 2*b**2*c*d**2*e*x**4 + b 
**2*c*d*e**2*x**6 + b*c**2*d**3*x**4 + b*c**2*d**2*e*x**6),x)*a**3*c**3...